Not recommended - wood isn't particularly compatible with tile...they expand and contract at different rates making the bond break or the tile.

Also, while not as likely on a vertical wall, stuff will get splashed on the backsplash, and with no grout in the joints, could leak back into the wall and you'd never get it out and clean again. Plan for at least 1/16" grout line.

The existing backsplash probably also has cooking grease from long exposure in the kitchen, also a bond breaker. It really needs to come out. If the walls are drywall, you could just cut it out and install new. The tile could be installed on drywall, or if you want, cement board. Thinset (dry powder comes in a bag) is the least expensive and best thing to hold the tile on the wall. Probably a lightly modified thinset like Versabond (available at HD). Mastic (stuff premixed in a bucket) probably could be used, but costs 4x as much and can't be used if the tile you choose are too big (it has to have some edges so it can dry...can't do that with a big tile and it will stay soft like in the bucket for months).
Check out www.johnbridge.com for help on tiling.